Recent Progress in Langlands Functoriality Conference, June 17-28, 2002, Luminy, France

为了承认朗兰兹计划内的最新进展并在此基础上再接再厉，CIRM已批准于2002年6月17日至28日召开一次具有上述标题的会议。组织者之一的研究人员认为，这样的会议将有助于推动任何出席的年轻研究人员的职业生涯。我们提议10名初级人员获得资金，以支持他们参加这些活动。申请的数额将足以支付旅费和当地费用。在管理这些基金时，我们将尽一切努力支持那些没有其他资金来源的人。这次会议的重点是综述被称为朗兰兹计划的数学分支的最新成果。这个项目的基本理念是，在三个看似不同的数学领域：代数几何、数论和调和分析之间应该有深层次的，而且在某种意义上，应该有基本的联系。Shimura-Taniyama猜想假定椭圆曲线理论(几何对象)和某些类型的自同构形式(具有数论应用的分析对象)等价性，只是朗兰兹程序所期望产生的一个‘简单’(在程序的上下文中是简单的)例子。通过证明这一猜想，Andrew Wiles给出了费马最后定理的一个长期寻求的证明，即对于大于2的整数，永远不存在n次方和为另一个非零整数的n次方的两个非零整数。朗兰兹计划中更一般的猜想可以用一些更专业的语言来表述，但与费马问题有一个相似之处，那就是证明比陈述要难得多。在过去的几年里，朗兰兹计划取得了一系列令人瞩目的突破。怀尔斯的工作是这些令人兴奋的结果的许多例子之一。正如提案中概述的那样，这是该计划历史上的关键时刻，因为必须找到新的想法来建立这种最近的势头。因此，为相对较新的研究人员提供资金参加高级别会议，例如由CIRM赞助的会议，将极大地服务于该领域和更大的数学界。正是通过这样的发展，数学在我们的社会中继续发挥着关键作用，无论是在文化上还是在技术上。

To recognize and build upon recent advances within the Langlands program, a conference with the above title has been approved by CIRM for June 17-28, 2002. The investigators, who are among the organizers, feel that such a conference will serve to advance the careers of any young researchers who are in attendance. We are proposing that ten junior people receive funds to support their participation in these proceedings. The amount requested will be sufficient for travel as well as local expenses. In administering these funds we shall make every effort to support those who have no other means of funding.The focus of this conference is to survey recent results within a branch of mathematics referred to as the Langlands program. The underlying philosophy of this program is that there should be deep, and in some sense, fundamental connections between three seemingly disparate mathematical fields: Algebraic Geometry, Number Theory, and Harmonic Analysis. The Shimura-Taniyama conjecture, which posited the equivalence of the theory of elliptic curves (geometric objects) and a certain classes of automorphic forms (analytic objects with number theoretic applications) is just a ``simple'' (that is simple within the context of the program) example of what the Langlands program is expected to produce. By proving this conjecture, Andrew Wiles delivered a long sought proof of Fermat's Last theorem, that for integers larger than 2, there is never a choice of two non-zero integers whose n-th powers sum to the n-th power of another non-zero integer. More general conjectures within the Langlands program can be stated with some more technical language, but have a similarity to the Fermat problem in that the proofs are much harder than the statements. The last several years have witnessed a series of remarkable breakthroughs in the Langlands program. Wiles's work is one of many examples of these exciting results. As outlined in the proposal, this is critical point in the history of the program, in that new ideas must be found to build upon this recent momentum. Thus, providing funding for relatively new researchers to attend a high level conference, such as the one being sponsored by CIRM, will serve the field, and the greater mathematical community greatly. It is through such development that mathematics continues to play a critical role in our society, both culturally and technologically.